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MONOTONICITY AND LOGARITHMIC CONVEXITY OF THREE FUNCTIONS INVOLVING EXPONENTIAL FUNCTION

Feng Qi|Bai-Ni Guo|Ai-Qi Liu
한국수학교육학회|2008.11|이론수학과 교직수학 (Theoretical Mathematics and Pedagogical Mathematics)|제15권 제4호|pp. 387-392

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참고문헌

키워드

inequalitymonotonicityextensionalternative prooflogarithmic convexityexponential func- tionHermite-Hadamard's integral inequality
주제분류수학

초록

In this note, an alternative proof and extensions are provided for the following conclusions in [6, Theorem 1 and Theorem 3]: The functions 1 x2 ¡ e¡x (1¡e¡x)2 and 1 t ¡ 1 et¡1 are decreasing in (0;1) and the function t eat¡e(a¡1)t for a 2 R and t 2 (0;1) is logarithmically concave.

Abstract

In this note, an alternative proof and extensions are provided for the following conclusions in [6, Theorem 1 and Theorem 3]: The functions 1 x2 ¡ e¡x (1¡e¡x)2 and 1 t ¡ 1 et¡1 are decreasing in (0;1) and the function t eat¡e(a¡1)t for a 2 R and t 2 (0;1) is logarithmically concave.

참고문헌 (11)

  1. Ch.-P. Chen, "Best constant in an inequality connected with exponential functions", Octogon Math. Mag, 12(2), pp.736~737 (2004)
  2. P.N. de Souza, "Berkeley Problems in Mathematics" (2001), Springer
  3. S. Guo, "A class of completely monotonic functions related to the remainder of Binet's formula with applications", Tamsui Oxf. J. Math. Sci, 25(1) (2009)
  4. B.-N. Guo, "Generalization of Bernoulli polynomials. Internat", J. Math. Ed. Sci. Tech, 33(3), pp.428~431 (2002)
  5. J.-Ch. Kuang, "Changyong Budengshi (Applied Inequalities)" (2004), Shandong Science and Technology Press
  6. A.-Q. Liu, "Monotonicity and logarithmic concavity of two functions involving exponential function", Internat. J. Math. Ed. Sci. Tech, 39(5), pp.686~691 (2008)
  7. F. Qi, "A monotonicity result of a function involving the exponential function and an application", RGMIA Res. Rep. Coll, 7(3), pp.507~509 (2004)
  8. F. Qi, "Generalisation of Bernoulli polynomials", RGMIA Res. Rep. Coll, 4(4), pp.691~695 (2001)
  9. F. Qi, "Generalizations and refinements of Hermite-Hadamard's inequality", Rocky Mountain J. Math, 35(1), pp.235~251 (2002)
  10. F. Qi, "Comparisons of two integral inequalities with Hermite-Hadamard-Jensen's integral inequality", Internat. J. Appl. Math. Sci, 3(1), pp.83~88 (2006)
  11. Y.-D. Wu, "The best constant for an inequality", Octogon Math. Mag, 12(1), pp.139~141 (2004)

관련 논문AI추천

THE LOG-CONVEXITY OF ANOTHER CLASS OF ONE-PARAMETER MEANS AND ITS APPLICATIONSZhen-Hang Yang대한수학회 · 대한수학회보2012 · 제49권 제1호COMPLETELY MONOTONIC FUNCTIONS RELATING TO THE GAMMA AND PSI FUNCTIONSX. Li장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2009 · 제12권 제1호SOME LOGARITHMICALLY COMPLETELY MONOTONIC FUNCTIONS RELATED TO THE GAMMA FUNCTIONBai-Ni Guo, Feng Qi대한수학회 · 대한수학회지2010 · 제47권 제6호COMPLETE MONOTONICITY OF A DIFFERENCE BETWEEN THE EXPONENTIAL AND TRIGAMMA FUNCTIONSFeng Qi, Xiao-Jing Zhang한국수학교육학회 · 이론수학과 교직수학2014 · 제21권 제2호A GENERALIZATION OF ALZER INEQUALITY RELATED TO EXPONENTIAL FUNCTIONL. Ying, W-Y Cui장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2015 · 제18권 제3호TWO NEW PROOFS OF THE COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE PSI FUNCTIONBai-Ni Guo, Feng Qi대한수학회 · 대한수학회보2010 · 제47권 제1호

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